properties of Riemann xi function

The Riemann xi function, defined by

$\xi (s):={\displaystyle \frac{s}{2}}(s-1){\pi }^{-\frac{s}{2}}\mathrm{\Gamma }\left({\displaystyle \frac{s}{2}}\right)\zeta (s),$

is an entire function having as zeros the nonreal zeros of the Riemann zeta function $\zeta$ and only them.

The modulus of the xi function is strictly increasing along every horizontal half-line lying in any open right half-plane that contains no xi zeros.  As well, the modulus decreases strictly along every horizontal half-line in any zero-free, open left half-plane.

Taking into account the functional equation

$$\xi (1-s)=\xi (s)$$

it follows the reformulation of the Riemann hypothesis:

Theorem.  The following three statements are equivalent.

(i). If $t$ is any fixed real number, then $|\xi (\sigma +it)|$ is increasing for  .

(ii). If $t$ is any fixed real number, then $|\xi (\sigma +it)|$ is decreasing for  .

(iii). The Riemann hypothesis is true.

Title	properties of Riemann xi function
Canonical name	PropertiesOfRiemannXiFunction
Date of creation	2013-03-22 19:35:25
Last modified on	2013-03-22 19:35:25
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Result
Classification	msc 11M06
Related topic	RobinsTheorem
Related topic	ExtraordinaryNumber